Question

Two strings of same material are stretched to thw the same tension .if their radii ate in the ratio 1:2 then respective wave velocities in them will be in ratio

Answer 1

v=√T/u. u- linear mass or

mass per unit

u=m/l length.

=πr^2l*density/l

=πr^2*density

r1=2r2 u2=1/2u1

u1:u2=2:1

Answer 2

The formula required is V = √(T / u)

where V = wave velocity ; T = tension acting ; u = linear mass or mass per unit length.

Thus, u = m / l (l = length of the string)

We know, d = m / v

where d = density ; m = mass ; v = volume.

Or, m = d.v

Then, u = d.v / l

Lets assume r₁ and r₂ are respective radii of the strings.

Given, r₂ = 2.r₁

Volume of string 1, v₁ = π.r₁².l₁

Therefore, u₁ = d.v₁ / l₁ = d.π.r₁²

Volume of string 2, v₂ = π.r₂².l₂

Therefore, u₂ = d.v₂ / l₂ = d.π.r₂²

As both the strings are made of same material, in both cases density(d) is same.

Given, same amount of tension is acting;

Thus, V₁ / V₂ = √(T / u₁) / √(T / u₂)

or, V₁ / V₂ = √(u₂ / u₁) = √(d.π.r₂² / d.π.r₁²) = √(r₂² / r₁²) = r₂ / r₁

or, V₁ / V₂ = 2.r₁ / r₁ = 2

Therefore, V₁ : V₂ = 2 : 1